Quantum repeater from quantum analog-digital interconverter

ABSTRACT

Quantum repeater systems and apparatus for quantum communication. In one aspect, a system includes a quantum signal receiver configured to receive a quantum field signal; a quantum signal converter configured to: sample quantum analog signals from a quantum field signal received by the quantum signal receiver; encode sampled quantum analog signals as corresponding digital quantum information in one or more qudits, comprising applying a hybrid analog-digital encoding operation to each quantum analog signal and a qudit in an initial state; decode digital quantum information stored in the one or more qudits as a recovered quantum field signal, comprising applying a hybrid digital-analog decoding operation to each qudit and a quantum analog register in an initial state; a quantum memory comprising qudits and configured to store digital quantum information encoded by the quantum signal converter; and a quantum signal transmitter configured to transmit the recovered quantum field signal.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a continuation of U.S. patent application Ser. No.17/975,313, filed on Oct. 27, 2022, which is a continuation of U.S.patent application Ser. No. 17/063,492, filed on Oct. 5, 2020, whichclaims the benefit of the filing date of U.S. Provisional ApplicationNo. 62/911,132, filed on Oct. 4, 2019. The contents of the priorapplications are incorporated herein by reference in their entirety.

TECHNICAL FIELD

This specification relates to quantum computing and quantumcommunication.

BACKGROUND

Quantum information includes any type of information carried by aquantum system. Quantum information can include quantum digitalinformation and quantum analog information.

The most basic unit of quantum digital information is the qubit. A qubitis a two-level quantum-mechanical system. In a classical system, a bitwould have to be in one level or the other. However, quantum mechanicsallows a qubit to be in a coherent superposition of both levels, aproperty which is fundamental to quantum mechanics and quantumcomputing.

Quantum analog information is carried by continuous-variable quantumsystems, e.g., quantum fields.

SUMMARY

This specification describes a quantum repeater system for quantumcommunication.

In general, one innovative aspect of the subject matter described inthis specification can be embodied in a quantum repeater system forquantum communication, the system comprising: a quantum signal receiverconfigured to receive a quantum field signal; a quantum signal converterconnected to the quantum signal receiver, wherein the quantum signalconverter is configured to: sample one or more quantum analog signalsfrom the quantum field signal received by the quantum signal receiver;encode sampled quantum analog signals as corresponding digital quantuminformation in one or more qudits, comprising, for each sampled quantumanalog signal, applying a hybrid analog-digital encoding operation tothe quantum analog signal and a qudit in an initial state; decodedigital quantum information stored in the one or more qudits as arecovered quantum field signal, comprising, for each of the one or morequdits, applying a hybrid digital-analog decoding operation to the quditand a quantum analog register in an initial state; a quantum memoryconnected to the quantum signal converter, wherein the quantum memorycomprises one or more qudits and is configured to store digital quantuminformation encoded by the quantum signal converter; and a quantumsignal transmitter connected to the quantum signal converter, whereinthe quantum signal transmitter is configured to transmit the recoveredquantum field signal.

The foregoing and other implementations can each optionally include oneor more of the following features, alone or in combination. In someimplementations the received quantum field signal comprises a Gottesman,Kitaev and Preskill (GKP) state; the one or more quantum analog signalssampled by the quantum signal converter comprise one or more GKP statesamples; and the recovered quantum field signal comprises a recoveredGKP state.

In some implementations the quantum memory is further configured toperform quantum error correction operations on digital quantuminformation stored in the quantum memory.

In some implementations the quantum error correction operations comprisesyndrome measurements, classical decoder operations and quantum errorcorrecting feedback operations.

In some implementations the quantum signal converter is furtherconfigured to decode error corrected digital quantum information storedin the one or more qudits as a recovered quantum field signal.

In some implementations the system further comprises multiple quantumsignal receivers, quantum signal converters, quantum memories, andquantum signal transmitters connected to form a quantum network.

In some implementations the hybrid digital-analog encoding operation andthe hybrid analog-digital decoding operation are based on a swapoperation comprising three adder operations.

In some implementations the hybrid digital-analog encoding operation andthe hybrid analog-digital decoding operation comprise: a first unitarytransformation comprising a canonical field momentum operator and aqudit field operator; multiple Fourier transformations; and a secondunitary transformation comprising a canonical field position operatorand the qudit field operator.

In some implementations applying the hybrid analog-digital encodingoperation to the quantum analog signal and a qudit in an initial statecomprises: applying the first unitary transformation to the quantumanalog signal and the initial state of the qudit to obtain a firstmodified quantum analog signal and a first evolved state of the quditsequentially applying two Fourier transformations to the first modifiedquantum analog signal to obtain a second modified quantum analog signal;applying a Fourier transformation to the first evolved state of thequdit to obtain a second evolved state of the qudit; applying the secondunitary transformation to the second modified quantum analog signal andthe second evolved state of the qudit to obtain a third modified quantumanalog signal and a third evolved state of the qudit; applying a Fouriertransformation to the third evolved state of the qudit to obtain afourth evolved state of the qudit; and applying the first unitarytransformation to the third modified quantum analog signal and thefourth evolved state of the qudit to obtain a fifth evolved state of thequdit, wherein providing the qudit in the evolved state as a quantumdigital encoding of the received quantum analog signal comprisesproviding the qudit in the fifth evolved state as a quantum digitalencoding of the received quantum analog signal.

In some implementations applying the hybrid digital-analog decodingoperation to the qudit and a quantum analog register in an initial statecomprises: sequentially applying two Fourier transformations to thequantum analog register in the initial state to obtain a first modifiedstate of the quantum analog register; applying a first unitarytransformation to the first modified state of the quantum analogregister and the fourth qudit to obtain a second modified state of thequantum analog register and a first evolved state of the fourth qudit,wherein the first unitary transformation comprises a canonical fieldmomentum operator and a qudit field operator; applying a Fouriertransformation to the first evolved state of the fourth qudit to obtaina second evolved state of the fourth qudit; applying a second unitarytransformation to the second modified state of the quantum analogregister and the second evolved state of the fourth qudit to obtain athird modified state of the quantum analog register and a third evolvedstate of the fourth qudit, wherein the second unitary transformationcomprises a canonical field position operator and the qudit fieldoperator; applying a Fourier transformation to the third evolved stateof the fourth qudit to obtain a fourth evolved state of the fourthqudit; sequentially applying two Fourier transformations to the thirdmodified state of the quantum analog register to obtain a fourthmodified state of the quantum analog register; and applying the firstunitary transformation to the fourth modified state of the quantumanalog register and the fourth evolved state of the fourth qudit toobtain a fifth modified state of the quantum analog register, whereinproviding the modified state of the quantum analog register as a quantumanalog encoding of the quantum digital information comprises providingthe fifth modified state of the quantum analog register as the quantumanalog encoding of the quantum digital information.

In general, another one innovative aspect of the subject matterdescribed in this specification can be embodied in a method forrepeating a quantum field signal, the method comprising: receiving aquantum field signal; sampling one or more quantum analog signals fromthe received quantum field signal; for each sampled quantum analogsignal: encoding the quantum analog signal as corresponding digitalquantum information, comprising applying a hybrid analog-digitalencoding operation to the quantum analog signal and a qudit in aninitial state; and storing the corresponding digital quantum informationin the qudit; generating a recovered quantum field signal, comprising:for each qudit storing corresponding digital quantum information,decoding the quantum digital information by applying a hybriddigital-analog decoding operation to the qudit and a quantum analogregister in an initial state; and combining the decoded quantum digitalinformation; and transmitting the recovered quantum field signal.

Other embodiments of this aspect include corresponding classical andquantum computer and communication systems, apparatus, and computerprograms recorded on one or more computer storage devices, eachconfigured to perform the actions of the methods. A system of one ormore classical and quantum computers and/or communication systems can beconfigured to perform particular operations or actions by virtue ofsoftware, firmware, hardware, or any combination thereof installed onthe system that in operation may cause the system to perform theactions. One or more computer programs can be configured to performparticular operations or actions by virtue of including instructionsthat, when executed by data processing apparatus, cause the apparatus toperform the actions.

The foregoing and other implementations can each optionally include oneor more of the following features, alone or in combination. In someimplementations the received quantum field signal comprises a Gottesman,Kitaev and Preskill (GKP) state; the one or more quantum analog signalssampled by the quantum signal converter comprise one or more GKP statesamples; and the recovered quantum field signal comprises a recoveredGKP state.

In some implementations storing the corresponding digital quantuminformation further comprises performing one or more rounds of quantumerror correction operations on digital quantum information stored in thequantum memory.

In some implementations the quantum error correction operations comprisesyndrome measurements, classical decoder operations and quantum errorcorrecting feedback operations.

In some implementations generating the recovered quantum field signalcomprises, for each qudit storing corresponding error corrected digitalquantum information, decoding the error corrected quantum digitalinformation by applying a hybrid digital-analog decoding operation tothe qudit storing error corrected digital quantum information and aquantum analog register in an initial state.

In some implementations the hybrid digital-analog encoding operation andthe hybrid analog-digital decoding operation are based on a swapoperation comprising three adder operations.

In some implementations the swap operation comprises: a first adderoperation applied to a first signal and a second signal; two sequentialFourier transformations applied to the second signal; a second adderoperation applied to the first signal and the second signal; twosequential Fourier transformations applied to the first signal; a thirdadder operation applied to the first signal and the second signal; andtwo sequential Fourier transformations applied to the second signal.

In some implementations the hybrid digital-analog encoding operation andthe hybrid analog-digital decoding operation comprise: a first unitarytransformation comprising a canonical field momentum operator and aqudit field operator; multiple Fourier transformations; and a secondunitary transformation comprising a canonical field position operatorand the qudit field operator.

In some implementations applying the hybrid analog-digital encodingoperation to the quantum analog signal and a qudit in an initial statecomprises: applying the first unitary transformation to the quantumanalog signal and the initial state of the qudit to obtain a firstmodified quantum analog signal and a first evolved state of the qudit;sequentially applying two Fourier transformations to the first modifiedquantum analog signal to obtain a second modified quantum analog signal;applying a Fourier transformation to the first evolved state of thequdit to obtain a second evolved state of the qudit; applying the secondunitary transformation to the second modified quantum analog signal andthe second evolved state of the qudit to obtain a third modified quantumanalog signal and a third evolved state of the qudit; applying a Fouriertransformation to the third evolved state of the qudit to obtain afourth evolved state of the qudit; and applying the first unitarytransformation to the third modified quantum analog signal and thefourth evolved state of the qudit to obtain a fifth evolved state of thequdit, wherein providing the qudit in the evolved state as a quantumdigital encoding of the received quantum analog signal comprisesproviding the qudit in the fifth evolved state as a quantum digitalencoding of the received quantum analog signal.

In some implementations applying the hybrid digital-analog decodingoperation to the qudit and a quantum analog register in an initial statecomprises: sequentially applying two Fourier transformations to thequantum analog register in the initial state to obtain a first modifiedstate of the quantum analog register; applying a first unitarytransformation to the first modified state of the quantum analogregister and the fourth qudit to obtain a second modified state of thequantum analog register and a first evolved state of the fourth qudit,wherein the first unitary transformation comprises a canonical fieldmomentum operator and a qudit field operator; applying a Fouriertransformation to the first evolved state of the fourth qudit to obtaina second evolved state of the fourth qudit; applying a second unitarytransformation to the second modified state of the quantum analogregister and the second evolved state of the fourth qudit to obtain athird modified state of the quantum analog register and a third evolvedstate of the fourth qudit, wherein the second unitary transformationcomprises a canonical field position operator and the qudit fieldoperator; applying a Fourier transformation to the third evolved stateof the fourth qudit to obtain a fourth evolved state of the fourthqudit; sequentially applying two Fourier transformations to the thirdmodified state of the quantum analog register to obtain a fourthmodified state of the quantum analog register; and applying the firstunitary transformation to the fourth modified state of the quantumanalog register and the fourth evolved state of the fourth qudit toobtain a fifth modified state of the quantum analog register, whereinproviding the modified state of the quantum analog register as a quantumanalog encoding of the quantum digital information comprises providingthe fifth modified state of the quantum analog register as the quantumanalog encoding of the quantum digital information.

The details of one or more implementations of the subject matter of thisspecification are set forth in the accompanying drawings and thedescription below. Other features, aspects, and advantages of thesubject matter will become apparent from the description, the drawings,and the claims.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a block diagram of an example quantum repeater system.

FIG. 2 is a flow diagram of an example process for repeating a quantumfield signal.

FIG. 3 is a flow diagram of an example process for generating a quantumdigital encoding of a quantum analog signal.

FIG. 4 shows an example swap operation.

FIG. 5 shows an example hybrid analog-digital encoding operation.

FIG. 6 is a flow diagram of an example process for generating a quantumanalog encoding of quantum digital information stored in a qudit.

FIG. 7 is a flow diagram of an example process for generating aGottesman-Kitaev-Preskill quantum state.

FIG. 8 is a flow diagram of an example process for converting aGottesman-Kitaev-Preskill quantum state to quantum digital information.

DETAILED DESCRIPTION

This specification describes a quantum repeater system for receiving andtransmitting quantum field signals. The quantum repeater system uses aquantum analog-digital interconverter that implements a hybridanalog-digital conversion operation to encode received quantum fieldsignals as digital quantum information. The digital quantum informationis stored in memory and can undergo one or more rounds of quantum errorcorrection. The quantum analog-digital interconverter can furtherimplement a hybrid digital-analog conversion operation to decode storeddigital quantum information as a recovered quantum field signal. In somecases the quantum analog-digital interconverter can encode, errorcorrect, and decode Gottesman-Kitaev-Preskill (GKP) quantum states.

FIG. 1 is a block diagram of an example quantum repeater system 100 forquantum communication. The system 100 is an example of a systemimplemented as computer programs on one or more classical and quantumcomputing devices in one or more locations, in which the systems,components, and techniques described below can be implemented.

The example system 100 includes a quantum signal receiver 102, a signalconverter 104, and a quantum signal transmitter 106. The example system100 is configured to receive as input a quantum field signal, e.g.,quantum field signal 112, and process the received quantum signal togenerate as output a recovered version of the quantum field signal,e.g., recovered quantum field signal 114. As described in more detailbelow, in some implementations the recovered quantum field signal can bean error corrected version of the received quantum field signal.

The quantum signal receiver 102 is configured to receive quantum fieldsignals. In some implementations the received quantum field signals caninclude Gottesman-Kitaev-Preskill (GKP) quantum states. The quantumsignal converter 104 is connected to the quantum signal receiver 102 andis configured to sample one or more quantum analog signals from quantumfield signals received by the quantum signal receiver 102. For example,the quantum signal converter 104 can be configured to perform operationsdescribed below with reference to step 302 of FIG. 3 . Inimplementations where the quantum signal receiver 102 receives a GKPstate, the quantum analog signals sampled by the quantum signalconverter can be GKP state samples. GKP quantum states are described inmore detail below with reference to FIG. 7 .

The quantum signal converter 104 is further configured to encode sampledquantum analog signals as corresponding digital quantum information inone or more qudits, e.g., stored in quantum memory 110. The quantumsignal converter 104 encodes a sampled quantum analog signal ascorresponding digital quantum information by applying a hybridanalog-digital encoding operation to the quantum analog signal and aqudit in an initial state using a quantum analog-digital interconverter108. An example hybrid analog-digital encoding operation is describedbelow with reference to FIG. 5 . An example process that the quantumanalog-digital interconverter 108 can be configured to perform forapplying a hybrid analog-digital encoding operation to a quantum analogsignal and a qudit in an initial state is described below with referenceto FIGS. 3-5 . An example process for applying a hybrid analog-digitalencoding operation to a GKP state sample and a qudit in an initial stateis described below with reference to FIG. 8 .

The quantum signal converter 104 includes or is connected to a quantummemory 110. As described above, the quantum memory 110 includes one ormore qudits (each including one or more qubits) and is configured tostore digital quantum information encoded by the quantum signalconverter 104. In some implementations the quantum memory 110 isconfigured to perform error correction operations on digital informationthat it stores. For example, the quantum memory 110 can be configured toperform one or more quantum error correction codes. Performing an errorcorrection code can include performing syndrome measurements on qubitsstored in the memory to determine whether the qubits have been corruptedor not, and applying classical decoders to infer a likely error thatoccurred. A quantum error correcting feedback operation to correct adetected error on a (logical) qubit can then be performed. Examplequantum error correction codes that the memory 110 can be configured toimplement include stabilizer codes, codes whose encoding and decodingoperations are not necessarily operations in the Clifford group, orapproximate error correcting codes.

The quantum signal converter 104 is further configured to decode (errorcorrected or non-error corrected) digital quantum information stored inthe one or more qudits in the quantum memory 110 as a recovered quantumfield signal. The quantum signal converter 104 decodes digital quantuminformation as a corresponding quantum field signal by applying a hybriddigital-analog encoding operation to the one or more qudits storing thedigital quantum information and a quantum analog signal in an initialstate using the quantum analog-digital interconverter 108. An examplehybrid digital-analog encoding operation and an example process that thequantum analog-digital interconverter 108 can be configured to performfor applying a hybrid digital-analog encoding operation to one or morequdits and a quantum analog signal in an initial state is describedbelow with reference to FIG. 6 . An example process for applying ahybrid digital-analog encoding operation to one or more qudits and aquantum analog signal in an initial state to produce a corresponding GKPstate is described below with reference to FIG. 7 .

The quantum signal transmitter 106 is connected to the quantum signalconverter 104 and is configured to transmit recovered quantum fieldsignals, e.g., recovered quantum field signal 114 as system output.

For convenience, the example quantum repeater system 100 shown in FIG. 1includes one quantum signal receiver, quantum signal converter, andquantum signal transmitter. However, in some implementations the systemcan include multiple quantum signal receivers, quantum signalconverters, and quantum signal transmitters, e.g., multiple copies ofsystem 100, connected to form a quantum network.

For purposes of transmitting information to the quantum signal receiver102 or from the quantum signal transmitter 106, one or more transmissionmedia can be used such as, but not limited to, free space optics,optical fibers, optical cavities, or any other medium where a quantumfield can be present (e.g., phonons in a solid material). In animplementation, the one or more qudits (e.g., one or more qubits) of thequantum signal converter 104 and/or the quantum memory 110 can includesuperconducting qudits (e.g., superconducting qubits). In general,superconducting qubits typically operate in the microwave regime.Accordingly, the quantum signal receiver 102 can convert receivedsignals from an optical signal to a microwave signal. Similarly, thequantum signal transmitter 106 can convert microwave signals to opticalsignals. Exemplary devices that can be used for the quantum signalreceiver 102 and transmitter 106 thus can include, but are not limitedto, quantum optomechanical devices, such as described, e.g., in“Microwave-to-optics conversion using a mechanical oscillator in itsquantum ground state” by M. Forsch et al., arXiv e-prints p.arXiv:1812.07588v1 (December 2018), or electro-optical devices, such asresonant whispering gallery mode resonators as described, e.g., in“Efficient microwave to optical photon conversion; an electro-opticalrealization” by A. Rueda et al., Optica, Vol. 3, No. 6, 597-604 (June2016).

The quantum signal converter 104 and/or the quantum memory 110 can bepart of a quantum computer (QC). In the exemplary implementation inwhich the qudits (or qubits) are superconducting elements, the quantumcomputer including the quantum signal converter 104 and/or the quantummemory 110 can be contained in a cryostat or other refrigeration devicethat includes electrical and thermal shielding and that can be operatedat or below the critical temperature of the superconducting materialsforming the qudits (or qubits). In certain implementations, the QC caninclude classical control electronics, static analog registers(cavities/resonators, among other types of registers), and internalinterconnects that relays the quantum field signal within the QC. Thequantum field signal, upon reaching the part of the QC used foranalog-digital conversion, can be sampled using, e.g., a form of static(as opposed to flying) oscillators. For instance, the sampling can beperformed using a set of superconducting LC circuit or set of resonantcavities. A set of superconducting qudits (or qubits) can be tunablycoupled to this latter set of analog qumodes. By tuning the couplingover time using control pulses sent from a set of control electronics,the quantum analog to digital operations can be executed. Subsequently,error correction can be applied.

FIG. 2 is a flow diagram of an example process 200 for repeating aquantum field signal. For convenience, the process 200 will be describedas being performed by a system of one or more classical and quantumcomputing devices located in one or more locations. For example, thequantum repeater system 100 of FIG. 1 can be configured to perform theexample process 200.

The system receives a quantum field signal (step 202). In someimplementations the receive quantum field signal can be a GKP state.

The system samples one or more quantum analog signals from the receivedquantum field signal (step 204). In cases where the received quantumfield is a GKP state, the one or more quantum analog signals can includeone or more GKP state samples. Sampling one or more quantum analogsignals from a received quantum field signal is described in more detailbelow with reference to step 302 of FIG. 3 .

For each sampled quantum analog signal, the system encodes the quantumanalog signal as corresponding digital quantum information (step 206)and stores the corresponding digital quantum information in a qudit(step 208). Encoding the quantum analog signal as corresponding digitalquantum information includes applying a hybrid analog-digital encodingoperation to the quantum analog signal and the qudit in an initialstate, as described in more detail below with reference to step 304 ofFIG. 3 .

In some implementations the system can perform one or more rounds ofquantum error correction on the stored qudits in order to keep thequantum state stored in memory error-free while waiting for next stepsin transmission.

The system generates a recovered quantum field signal by, for each quditstoring corresponding digital quantum information, decoding the quantumdigital information by applying a hybrid digital-analog decodingoperation to the qudit and a quantum analog register in an initial state(step 210). In implementations where the system performs one or morerounds of quantum error correction on the stored qudits, the system cangenerate a recovered quantum field signal by decoding the errorcorrected quantum digital information. Decoding quantum digitalinformation by applying a hybrid digital-analog decoding operation to aqudit and a quantum analog register in an initial state is described inmore detail below with reference to step 604 of FIG. 6 .

The system combines the decoded quantum digital information andtransmits the recovered quantum field signal (step 212).

FIG. 3 is a flow diagram of an example process 300 for generating aquantum digital encoding of a quantum analog signal. For convenience,the process 300 will be described as being performed by a system of oneor more classical and quantum computing devices located in one or morelocations. For example, the quantum repeater system 100 of FIG. 1 can beconfigured to perform the example process 300.

The system obtains a quantum analog signal (step 302). The quantumanalog signal can include a quantum mode of a quantum field and aquantum mode amplitude sampled from an interval of space, frequency, ora general window function profile of the quantum field. In someimplementations the quantum mode amplitude can be an average fieldamplitude value determined according to a predetermined window function,e.g., a wavelet, and a quantum field operator corresponding to thequantum field.

In some implementations the system can obtain the quantum analog signalby sampling the quantum mode and quantum mode amplitude of the quantumfield, e.g., using a resonator coupled to the quantum field. In theseimplementations the system can store the sampled quantum mode andquantum mode amplitude in an analog register, e.g., in the resonatorcoupled to the quantum field.

To sample the quantum mode and quantum mode amplitude of the quantumfield, the system can transfer quantum information from the quantumfield onto a quantum mode (a continuous-variable quantum analog degreeof freedom or memory, e.g. a quantum harmonic oscillator) throughapplication of an analog swap operation to the quantum analog signal andthe quantum mode. This can include coupling the two quantum degrees offreedom via a form of controllable coupling in order to convert a givensample contained in the “flying” memory (e.g. electromagnetic signalmoving at the speed of light) onto a stationary quantum analog memoryelement, e.g., on a chip. The analog swap operation can be applied byimplementing a unitary operator

$= e^{i\frac{\pi}{2}{({{{\hat{a}}_{j}^{\dagger}{\hat{a}}_{k}} + {{\hat{a}}_{j}{\hat{a}}_{k}^{\dagger}}})}}$

where â_(k) and â_(j) represent photon annihilation operators of thek-th and j-th quantum mode respectively. In some implementations theindex j can label a sample subspace of the quantum field, and the indexk can label the stationary quantum mode on the chip. This unitary swapis the result of an evolution under photon exchange interaction,commonly occurring in beam splitters in optical systems, or whenever twobosonic quantum modes are in resonance with one another (i.e. stronglycoupled).

The system applies a hybrid analog-digital encoding operation to thequantum analog signal and a qudit in an initial state to obtain anevolved state of the qudit (step 304). The qudit includes a d=2^(N)dimensional quantum register represented by N qubits, where N isselected based on a predetermined target encoding precision. The quditcan be prepared in an arbitrary initial state. During the process 300the state of the qudit will be transferred to the quantum analog signal,which enables simultaneous emission and receiving of quantuminformation. In the case of example process 300, the quantum analogsignal is being encoded as quantum digital information and therefore thetransfer of the initial state of the qudit to the quantum analog signalis not of primary importance. However, for certain initial qudit states,some operations of the example process 300 can be eliminated. Forexample, if the qudit is prepared in a |0>state, a first adder operationin the swap operation described below can be omitted since applying anadder operation to the |0>state leaves the system invariant and thus theoperation can be omitted.

The hybrid analog-digital encoding operation is based on a swapoperation that operates on two signals—a first signal and a secondsignal—and includes multiple adder operations. In some implementationsthe multiple adder operations can include three adder operations. Theswap operation can also include multiple quantum Fouriertransformations. For example, the swap operation can include a firstadder operation applied to a first signal and a second signal, twosequential Fourier transformations applied to the second signal, asecond adder operation applied to the first signal and the secondsignal, two sequential Fourier transformations applied to the firstsignal, a third adder operation applied to the first signal and thesecond signal, and two sequential Fourier transformations applied to thesecond signal.

The swap operation can be an analog swap operation that operates on afirst quantum analog signal and a second quantum analog signal and swapsinformation stored in the first quantum analog signal and the secondquantum analog signal. In this case the above described first adderoperation and third adder operation represent a unitary transformationU₁=e^(i{circumflex over (ϕ)}) ¹ ^({circumflex over (π)}) ² that includesa canonical field position operator {circumflex over (ϕ)}₁ for the firstquantum analog signal and a canonical field momentum operator{circumflex over (ϕ)}₂ for the second quantum analog signal. The secondadder operation represents a unitary transformationU₂=e^(i{circumflex over (ϕ)}) ¹ ^({circumflex over (π)}) ² comprising acanonical field momentum operator {circumflex over (ϕ)}₂ for the firstquantum analog signal and a canonical field position operator 432 forthe second quantum analog signal. However, in practice, a more efficientimplementation of an analog swap operation can be achieved throughevolution under photon exchange interaction, as described above.

Alternatively, the swap operation can be a digital swap operation thatoperates on a first quantum digital signal and a second quantum digitalsignal and swaps information stored in the first quantum digital signaland the second quantum digital signal. In this case the above describedfirst adder operation, second adder operation and third adder operationrepresent a unitary transformation U=e^(iĴ) ¹ ^(Ĵ) ² that includes afirst qudit clock operator generator Ĵ₁ for the first quantum digitalsignal and a second qudit clock operator generator Ĵ₂ for the secondquantum digital signal.

FIG. 4 shows an example swap operation 400 applied to a first signal 402a and a second signal 402 b. As described above, the first signal 402 aand second signal 402 b can both be quantum analog signals or both bequantum digital signals. If the first signal 402 a and second signal 402b are quantum analog signals, the adder operations 404, 408 and 412represent the unitary transformations given in the legend 416. If thefirst signal 402 a and second signal 402 b are quantum digital signals,the adder operations 404, 408 and 412 represent the unitarytransformations given in the legend 418.

During application of the example swap operation 400, a first adderoperation 404 is applied to the first signal 402 a and the second signal402 b. Two quantum Fourier transformations 406 a, 406 b are thensequentially applied to the second signal 402 b. In practicalimplementations, sequential application of two quantum Fouriertransforms to an analog quantum signal can be achieved through a singleoperation that includes application of a pi pulse to the analog quantumsignal, e.g. Û=F_(j) ²=e^(iπ(â) ^(j) ^(†â) ^(j) ⁾. Application of the pipulse represents an evolution under a quantum harmonic oscillatorHamiltonian for an angle (i.e., time multiplied by angular frequency)π.

A second adder operation 408 is then applied to the first signal 402 aand the second signal 402 b. Two quantum Fourier transformations 410 a,410 b are then sequentially applied to the first signal 402 a. Again, inpractical implementations sequential application of the two quantumFourier transforms can be achieved through application of a pi pulse tothe first signal 402 a.

A third adder operation 412 is then applied to the first signal 402 aand the second signal 402 b. The third adder operation is the same asthe first adder operation 404. Two quantum Fourier transformations 414a, 414 b are then sequentially applied to the second signal 402 b.Again, in practical implementations sequential application of the twoquantum Fourier transforms can be achieved through application of a pipulse to the second analog quantum signal 402 b.

Returning to FIG. 3 , the hybrid analog-digital encoding operation thatis based on the above described swap operation includes a first unitarytransformation that includes a canonical field momentum operator and aqudit field operator. The qudit field operator is given by a linearcombination of qudit clock operator generators Ĵ_(d=2) _(N) =Σ_(N=1)^(N)2^(n-2)(Î₂ ^((n))Z₂ ^((n))), where Î₂ ^((n)) represents a 2×2identity operator acting on qubit n and Z₂ ^((n)) represents a Pauli Zoperator acting on qubit n, and identity operators. For example, thequdit field operator can be given by

$\Phi_{d} = {{\frac{\left( {b - a} \right)}{\left( {d - 1} \right)}{\overset{\hat{}}{J}}_{d}} + {a{\hat{I}}_{d}}}$

where id represents a d×d identity operator and [a, b] represents aquantum analog sampling interval where a and b are tunable parameterswhich can be tuned to sample from different values of position.

The hybrid analog-digital encoding operation also includes multiplequantum Fourier transformations, and a second unitary transformationthat includes a canonical field position operator and the qudit fieldoperator. Because the qudit includes a d=2^(N) dimensional quantumregister represented by N qubits, applications of the first unitarytransformation and the second unitary transformation to states of thequdit involves applying corresponding qubit transformations torespective states of the N qubits.

The hybrid analog-digital encoding operation is approximately equivalentto the swap operation, e.g., up to a given fidelity, precision and/orrange limits determined by the dimension of the qudit (number ofqubits).

FIG. 5 shows an example hybrid analog-digital encoding operation 500.The example hybrid analog-digital encoding operation 500 is described asbeing applied to a quantum analog signal 502 and a qudit 504 prepared aninitial state, where the qudit represents a d=dimensional quantumregister that includes N qubits. However, the example hybridanalog-digital encoding operation 500 could also be applied directly tothe quantum analog signal 502 and the N qubits, i.e., the quantum analogsignal 502 could also be coupled directly to the N qubits.

During application of the example hybrid analog-digital encodingoperation 500, a first unitary transformation 506 is applied to thequantum analog signal 502 and the initial state of the qudit 504 toobtain a first modified quantum analog signal and a first evolved stateof the qudit. The first unitary transformation includes a canonicalfield position operator {circumflex over (Φ)}_(d) for the qudit 504 anda canonical field momentum operator {circumflex over (π)} for thequantum analog signal 502. That is, the first unitary transformation isgiven by U=e^(i{circumflex over (Φ)}) ^(d) ^({circumflex over (π)}).

Since the qudit represents a d=2^(N) dimensional quantum registerrepresented by N qubits, application of the first unitary transformation506 represents an evolution under multiple one-to-one interactionsbetween each of the N qubits and the stationary quantum analog signal502. That is, the first unitary transformation 506 can represent a totalevolution under each of the one-to-one interactions, e.g., a product ofindividual unitary transformations.

Two quantum Fourier transformations 508 a, 508 b are then sequentiallyapplied to the first modified quantum analog signal to obtain a secondmodified quantum analog signal. As described above with reference toFIG. 4 , in practical implementations sequential application of twoquantum Fourier transforms to a quantum analog signal can be achievedthrough application of a pi pulse to the analog quantum signal.

A quantum Fourier transformation 510 is applied to the first evolvedstate of the qudit to obtain a second evolved state of the qudit. Asecond unitary transformation 512 is applied to the second modifiedquantum analog signal and the second evolved state of the qudit toobtain a third modified quantum analog signal and a third evolved stateof the qudit. The second unitary transformation includes a canonicalfield position operator {circumflex over (Φ)}_(d) for the qudit 504 anda canonical field position operator {circumflex over (ϕ)} for thequantum analog signal 502. That is, the second unitary transformation isgiven by U=e^(i{circumflex over (Φ)}) ^(d) ^({circumflex over (π)}).

A quantum Fourier transformation 514 is applied to the third evolvedstate of the qudit to obtain a fourth evolved state of the qudit.

The first unitary transformation 516 is then applied to the thirdmodified quantum analog signal and the fourth evolved state of the quditto obtain a fourth modified quantum analog signal and a fifth evolvedstate of the qudit. The fifth evolved state of the qudit can be providedas a quantum digital encoding 522 of the received quantum analog signal,as described below with reference to step 306 of FIG. 3 .

Application of the example hybrid analog-digital encoding operation 500can also include sequentially applying two quantum Fouriertransformations 518 a, 518 b to the fourth modified quantum analogsignal. Application of the two quantum Fourier transformations 518 a,518 b is not essential for the encoding process 300, however the twoquantum Fourier transformations 518 a, 518 b must be included in theexample hybrid analog-digital encoding operation 500 if the encodingoperation is to be a swap operation, i.e., if the example hybridanalog-digital encoding operation 500 is to be a reversible operation.

Returning to FIG. 3 , the system provides the qudit in the evolved stateas a quantum digital encoding of the received quantum analog signal(step 306). Alternatively or in addition, the system can store thequantum digital encoding of the received quantum analog signal inquantum memory.

In some implementations the system can discard one or more of the Nqubits to reduce the resolution of the quantum digital encoding of thereceived quantum analog signal when providing the qudit in the fifthevolved state as the quantum digital encoding of the received quantumanalog signal. This process is illustrated in FIG. 5 , where a firstnumber of the N qubits represented by the qudit 504 are provided as thequantum digital encoding 522 of the quantum analog signal 502, and asecond number of the N qubits represented by the qudit 504 are bufferqubits 520 and are discarded.

The example process 300 can be repeated to generating multiple quantumdigital encodings of respective quantum analog signals. For example, atstep 302 the system can receive multiple quantum analog signals whereeach of the multiple quantum analog signals includes a respectivequantum mode of a same quantum field, e.g., where the respective quantummodes of the same quantum field form a basis, and a respective quantummode amplitude sampled from an interval of the quantum field. In someimplementations the multiple quantum analog signals can include quantumanalog signals that include a same quantum mode and respective quantummode amplitudes sampled from different intervals of the quantum field,e.g., where the different sampling intervals of the quantum field areselected based on a Nyquist-Shannon sampling rate.

The system can then apply the hybrid analog-digital encoding operationto each of the multiple quantum analog signals and a qudit in an initialstate to obtain multiple qudits in respective evolved states as aquantum digital encoding of the multiple quantum analog signals. In thisexample, the provided quantum digital encodings of the received multiplequantum analog signals can form a quantum digital encoding of thequantum field.

In some implementations the system can sequentially sample and apply thehybrid analog-digital encoding operation to each of the multiple quantumanalog signals. In these implementations the system can apply a holdoperation to the analog quantum modes in memory during application ofeach hybrid analog-digital encoding operation.

FIG. 6 is a flow diagram of an example process 600 for generating aquantum analog encoding of quantum digital information stored in aqudit. For convenience, the process 400 will be described as beingperformed by a system of one or more classical and quantum computingdevices located in one or more locations. For example, the quantumrepeater system 100 of FIG. 1 can be configured to perform the exampleprocess 600.

The system obtains a qudit that stores quantum digital information (step602). The qudit includes a d=2^(N) dimensional quantum registerrepresented by N qubits. In some implementations N can be selected basedon a predetermined target encoding precision. For example, in some casesthe N qubits can include additional qubits, i.e., qubits that do notstore the quantum digital information that is to be encoded as a quantumanalog signal, to increase the resolution of the quantum analog encodingof the quantum digital information (to give more range in signal phasespace, as well as finer-grained precision/sharpness, i.e. a lowfine-grained precision state would seem blurry. By tuning the dimensionof the system, this range in phase space can be tuned. Phase space isthe space of position and momentum of each signal, depicted as input andoutput 302 and 322 in FIG. 3 ).

The system applies a hybrid digital-analog encoding operation to thequdit and a quantum analog register in an initial state to obtain amodified state of the quantum analog register (step 604). The initialstate of the quantum analog register can include one or more quantummodes, as described above with reference to FIG. 3 . In someimplementations the initial state can be a vacuum state or a thermalstate, however any state of known range in amplitude and momentum couldbe used. The use of states with unknown ranges in amplitude and momentumcould incur some dithering/aliasing effects, similar to classical undersampling effects. Therefore, if an initial state with amplitude andmomentum outside of a known range is used, a non-negligible probabilityof error can need to be tolerated.

The hybrid digital-analog encoding operation is based on the swapoperation described above with reference to FIGS. 3 and 4 , and forbrevity is not described again. In addition, application of the hybriddigital-analog encoding operation is the same as a reverse applicationof the hybrid analog-digital encoding operation (including quantumFourier transformations 518 a, 518 b) described above with reference toFIGS. 3 and 5 , since the example hybrid analog-digital encodingoperation illustrated in FIG. 3 is a swap operation and thereforereversible.

Therefore, applying the hybrid digital-analog swap operation to thequdit and the quantum analog register in the initial state includes:sequentially applying the Fourier transformations 518 a, 518 b of FIG. 5(or a pi pulse as described above) to the quantum analog register 502 inthe initial state to obtain a first modified state of the quantum analogregister. The first unitary transformation 516 is then applied to thefirst modified state of the quantum analog register and the qudit 504 toobtain a second modified state of the quantum analog register and afirst evolved state of the qudit. The Fourier transformation 514 is thenapplied to the first evolved state of the qudit to obtain a secondevolved state of the qudit. The second unitary transformation 512 isthen applied to the second modified state of the quantum analog registerand the second evolved state of the qudit to obtain a third modifiedstate of the quantum analog register and a third evolved state of thequdit. The Fourier transformation 510 is then applied to the thirdevolved state of the qudit to obtain a fourth evolved state of thequdit. The Fourier transformations 508 a, 508 b (or a pi pulse asdescribed above) are then sequentially applied to the third modifiedstate of the quantum analog register to obtain a fourth modified stateof the quantum analog register. The first unitary transformation 506 isthen applied to the fourth modified state of the quantum analog registerand the fourth evolved state of the qudit to obtain a fifth modifiedstate of the quantum analog register. The fifth modified state of thequantum analog register is then provided as a quantum analog encoding ofthe quantum digital information (step 606).

The example process 600 can be repeated to generating multiple quantumanalog encodings of respective quantum digital information stored inmultiple qudits. For example, at step 602 the system can receivemultiple qudits, where each qudit stores respective quantum digitalinformation. The system can then apply the hybrid digital-analog swapoperation to each qudit and a quantum analog register in an initialstate to obtain multiple modified states of quantum analog registers asa quantum analog encoding of the quantum digital information. In someimplementations the states of the quantum analog registers can becombined to produce a quantum field that encodes the information storedin the multiple qudits. For example, the quantum field can interact withthe quantum analog registers (analog memory quantum modes) in a similarway to that described above with reference to FIG. 3 —through swappinginteractions of the form

$= e^{i\frac{\pi}{2}{({{{\hat{a}}_{j}^{\dagger}{\hat{b}}_{k}} + {{\hat{a}}_{j}{\hat{b}}_{k}^{\dagger}}})}}$

where â_(j) represents the annihilation operator of memory quantum modej and {circumflex over (b)}_(k) represents the annihilation operator ofsmeared observable subsystem k (window of quantum field). An example ofa set of smeared observable subsystems is: {circumflex over(ϕ)}_(J)≡∫dxλ_(j)(x){circumflex over (ϕ)}(x) where λ_(j) representsL{circumflex over ( )}2-normalized window functions and Φ(x) representsthe quantum field amplitude at point x. The canonical conjugate of theseamplitude observables are {circumflex over(π)}_(j)≡∫dxλ_(j)(x){circumflex over (Π)}(x) with the same normalizedwindow function, and {circumflex over (Π)}(x) represents the quantumfield canonical conjugate to the amplitude at point x. The annihilationoperators are defined as

${\overset{\hat{}}{b}}_{j} = \frac{1}{\sqrt{2}}$

and the corresponding creation operator is the Hermitian conjugate.

FIG. 7 is a flow diagram of an example process 700 for generating atarget GKP quantum state. The target GKP state includes a series ofGaussian peaks of target width a and target tunable separation α√{squareroot over (π)}, α∈

embedded in a larger Gaussian envelope of target width 1/σ. Although inthe case of infinite squeezing (σ→0) the GKP state bases becomeorthogonal, in the case of finite squeezing, the approximate code statesare not orthogonal. The approximate GKP code states |0) and |1) can bedefined as

$\left. ❘0 \right\rangle \propto {\sum\limits_{t = {- \infty}}^{\infty}{\int{\left. e^{{- 2}\pi\sigma^{2}t^{2}}e^{{- {({q - {2t\sqrt{\pi}}})}^{2}}/{({2\sigma^{2}})}}❘q \right\rangle{dq}}}}$$\left. ❘1 \right\rangle \propto {\sum\limits_{t = {- \infty}}^{\infty}{\int{\left. e\frac{\pi{\sigma^{2}\left( {{2t} + 1} \right)}^{2}}{2}e^{{- {({q - {{({{2t} + 1})}\pi}})}^{2}}/{({2\sigma^{2}})}}❘q \right\rangle{{dq}.}}}}$

For convenience, the process 700 will be described as being performed bya system of one or more classical and quantum computing devices locatedin one or more locations. For example, the quantum repeater system 100of FIG. 1 can be configured to perform the example process 700.

The system obtains a fourth qudit in an initial state (step 702). Theinitial state of the fourth qudit includes a tensor product of a stateof a first qudit, a state of a second qudit, and a state of a thirdqudit. The state of the first qudit encodes the Gaussian envelope oftarget width 1/σ. The state of the first qudit can be represented by afirst Gaussian wavefunction. The state of the second qudit encodes thetarget separation α√{square root over (π)}. That is, the state of thesecond qudit includes logical information that determines a position ofthe Gaussian peaks of target width a. The state of the second qubit canbe a general superposition sate. The state of a third qudit encodes thetarget width a. The state of the third qudit can be represented by asecond Gaussian wavefunction.

The fourth qudit can include a d=2^(N) dimensional quantum registerrepresented by N qubits. For example, the first qudit can include afirst multiple of qubits, the second qudit can include a second multipleof qubits, and the third qudit can include a third multiple of qubits,where the first multiple added to the second multiple added to the thirdmultiple is equal to d=2^(N).

The merging of the first, second and third qudits to produce the fourthqudit can include labelling the qubits of the first, second and thirdqudits in order of precision, i.e., what power of the position value thequbit represents. For example, by appending two multi-qubit quditregisters, the original intra-sub-qudit ordering is maintained. However,the collection of all qubits are considered as forming onelarger-dimensional qudit. For the presently described GKP construction,three different qudits are merged. Each qudit determines a wavefunctionfor intervals of precision. For the low-precision, e.g., large-scalefeatures, the first qudit determines the shape of the larger Gaussianenvelope of width 1/σ. The second qudit represents mid-precision andcontains logical information including superpositions of differentpositions for the next tier's wavefunction (the higher precision). Thethird qudit represents high precision. The third qudit wavefunctiondetermines the Gaussian peaks of target width a (the finest graininformation). All three qudits put together as a tensor product andconsidered as a single qudit (fourth qudit) approximating a continuousone-dimensional quantum system result in a wavefunction that includes asuperposition (depending on logical information) of the Gaussianapproximate Dirac combs described above. The variances of the Gaussiansof the outer larger Gaussian envelope and Gaussian peaks can be tuned.In general it can be beneficial to keep these dual to each other, suchthat the Fourier transform of the wavefunction is also a Gaussianapproximate Dirac comb.

The system applies a hybrid digital-analog swap operation to the fourthqudit and a quantum analog register in an initial state to obtain amodified state of the quantum analog register (step 704). The hybriddigital-analog swap operation is based on a swap operation that includesmultiple adder operations. An example hybrid digital-analog swapoperation and an example process for applying a hybrid digital-analogswap operation to a qudit and a quantum analog register in an initialstate is described below with reference to FIGS. 4-6 .

The system provides the modified state of the quantum analog register asan approximate Gottesman-Kitaev-Preskill quantum state (step 706). Insome implementations the approximate Gottesman-Kitaev-Preskill quantumstate can be provided for use in a quantum computation or quantumcommunication protocol or method. For example, the approximateGottesman-Kitaev-Preskill quantum state can be transmitted to a quantumrepeater, where the quantum repeater can receive the transmittedGottesman-Kitaev-Preskill quantum state, perform analog error correctionoperations on the received Gottesman-Kitaev-Preskill quantum state toobtain an error corrected Gottesman-Kitaev-Preskill quantum state, andre-transmit the error corrected Gottesman-Kitaev-Preskill quantum state.

FIG. 8 is a flow diagram of an example process 800 for converting aGottesman-Kitaev-Preskill quantum state to quantum digital information.For convenience, the process 800 will be described as being performed bya system of one or more classical and quantum computing devices locatedin one or more locations. For example, the quantum repeater system 100of FIG. 1 can be configured to perform the example process 800.

The system obtains a quantum analog register in a GKP quantum state(step 802).

The system applies a hybrid analog-digital conversion operation to thequantum analog register and a qudit in an initial state to obtain anevolved state of the qudit (step 804). The hybrid analog-digital swapoperation is based on a swap operation that includes multiple adderoperations. An example hybrid analog-digital swap operation and anexample process for applying a hybrid digital-analog swap operation to aquantum analog register in a given state and a qudit in an initial stateis described below with reference to FIGS. 3-5 .

The system provides the qudit in the evolved state as a quantum digitaldecoding of the GKP quantum state (step 806).

Implementations of the digital and/or quantum subject matter and thedigital functional operations and quantum operations described in thisspecification can be implemented in digital electronic circuitry,suitable quantum circuitry or, more generally, quantum computationalsystems, in tangibly-embodied digital and/or quantum computer softwareor firmware, in digital and/or quantum computer hardware, including thestructures disclosed in this specification and their structuralequivalents, or in combinations of one or more of them. The term“quantum computational systems” can include, but is not limited to,quantum computers, quantum information processing systems, quantumcryptography systems, or quantum simulators.

Implementations of the digital and/or quantum subject matter describedin this specification can be implemented as one or more digital and/orquantum computer programs, i.e., one or more modules of digital and/orquantum computer program instructions encoded on a tangiblenon-transitory storage medium for execution by, or to control theoperation of, data processing apparatus. The digital and/or quantumcomputer storage medium can be a machine-readable storage device, amachine-readable storage substrate, a random or serial access memorydevice, one or more qubits, or a combination of one or more of them.Alternatively or in addition, the program instructions can be encoded onan artificially-generated propagated signal that is capable of encodingdigital and/or quantum information, e.g., a machine-generatedelectrical, optical, or electromagnetic signal, that is generated toencode digital and/or quantum information for transmission to suitablereceiver apparatus for execution by a data processing apparatus.

The terms quantum information and quantum data refer to information ordata that is carried by, held or stored in quantum systems, where thesmallest non-trivial system is a qubit, i.e., a system that defines theunit of quantum information. It is understood that the term “qubit”encompasses all quantum systems that can be suitably approximated as atwo-level system in the corresponding context. Such quantum systems caninclude multi-level systems, e.g., with two or more levels. By way ofexample, such systems can include atoms, electrons, photons, ions orsuperconducting qubits. In many implementations the computational basisstates are identified with the ground and first excited states, howeverit is understood that other setups where the computational states areidentified with higher level excited states are possible.

The term “data processing apparatus” refers to digital and/or quantumdata processing hardware and encompasses all kinds of apparatus,devices, and machines for processing digital and/or quantum data,including by way of example a programmable digital processor, aprogrammable quantum processor, a digital computer, a quantum computer,multiple digital and quantum processors or computers, and combinationsthereof. The apparatus can also be, or further include, special purposelogic circuitry, e.g., an FPGA (field programmable gate array), an ASIC(application-specific integrated circuit), or a quantum simulator, i.e.,a quantum data processing apparatus that is designed to simulate orproduce information about a specific quantum system. In particular, aquantum simulator is a special purpose quantum computer that does nothave the capability to perform universal quantum computation. Theapparatus can optionally include, in addition to hardware, code thatcreates an execution environment for digital and/or quantum computerprograms, e.g., code that constitutes processor firmware, a protocolstack, a database management system, an operating system, or acombination of one or more of them.

A digital computer program, which may also be referred to or describedas a program, software, a software application, a module, a softwaremodule, a script, or code, can be written in any form of programminglanguage, including compiled or interpreted languages, or declarative orprocedural languages, and it can be deployed in any form, including as astand-alone program or as a module, component, subroutine, or other unitsuitable for use in a digital computing environment. A quantum computerprogram, which may also be referred to or described as a program,software, a software application, a module, a software module, a script,or code, can be written in any form of programming language, includingcompiled or interpreted languages, or declarative or procedurallanguages, and translated into a suitable quantum programming language,or can be written in a quantum programming language, e.g., QCL orQuipper.

A digital and/or quantum computer program may, but need not, correspondto a file in a file system. A program can be stored in a portion of afile that holds other programs or data, e.g., one or more scripts storedin a markup language document, in a single file dedicated to the programin question, or in multiple coordinated files, e.g., files that storeone or more modules, sub-programs, or portions of code. A digital and/orquantum computer program can be deployed to be executed on one digitalor one quantum computer or on multiple digital and/or quantum computersthat are located at one site or distributed across multiple sites andinterconnected by a digital and/or quantum data communication network. Aquantum data communication network is understood to be a network thatcan transmit quantum data using quantum systems, e.g. qubits. Generally,a digital data communication network cannot transmit quantum data,however a quantum data communication network can transmit both quantumdata and digital data.

The processes and logic flows described in this specification can beperformed by one or more programmable digital and/or quantum computers,operating with one or more digital and/or quantum processors, asappropriate, executing one or more digital and/or quantum computerprograms to perform functions by operating on input digital and quantumdata and generating output. The processes and logic flows can also beperformed by, and apparatus can also be implemented as, special purposelogic circuitry, e.g., an FPGA or an ASIC, or a quantum simulator, or bya combination of special purpose logic circuitry or quantum simulatorsand one or more programmed digital and/or quantum computers.

For a system of one or more digital and/or quantum computers to be“configured to” perform particular operations or actions means that thesystem has installed on it software, firmware, hardware, or acombination of them that in operation cause the system to perform theoperations or actions. For one or more digital and/or quantum computerprograms to be configured to perform particular operations or actionsmeans that the one or more programs include instructions that, whenexecuted by digital and/or quantum data processing apparatus, cause theapparatus to perform the operations or actions. A quantum computer canreceive instructions from a digital computer that, when executed by thequantum computing apparatus, cause the apparatus to perform theoperations or actions.

Digital and/or quantum computers suitable for the execution of a digitaland/or quantum computer program can be based on general or specialpurpose digital and/or quantum processors or both, or any other kind ofcentral digital and/or quantum processing unit. Generally, a centraldigital and/or quantum processing unit will receive instructions anddigital and/or quantum data from a read-only memory, a random accessmemory, or quantum systems suitable for transmitting quantum data, e.g.photons, or combinations thereof.

The essential elements of a digital and/or quantum computer are acentral processing unit for performing or executing instructions and oneor more memory devices for storing instructions and digital and/orquantum data. The central processing unit and the memory can besupplemented by, or incorporated in, special purpose logic circuitry orquantum simulators. Generally, a digital and/or quantum computer willalso include, or be operatively coupled to receive digital and/orquantum data from or transfer digital and/or quantum data to, or both,one or more mass storage devices for storing digital and/or quantumdata, e.g., magnetic, magneto-optical disks, optical disks, or quantumsystems suitable for storing quantum information. However, a digitaland/or quantum computer need not have such devices.

Digital and/or quantum computer-readable media suitable for storingdigital and/or quantum computer program instructions and digital and/orquantum data include all forms of non-volatile digital and/or quantummemory, media and memory devices, including by way of examplesemiconductor memory devices, e.g., EPROM, EEPROM, and flash memorydevices; magnetic disks, e.g., internal hard disks or removable disks;magneto-optical disks; CD-ROM and DVD-ROM disks; and quantum systems,e.g., trapped atoms or electrons. It is understood that quantum memoriesare devices that can store quantum data for a long time with highfidelity and efficiency, e.g., light-matter interfaces where light isused for transmission and matter for storing and preserving the quantumfeatures of quantum data such as superposition or quantum coherence.

Control of the various systems described in this specification, orportions of them, can be implemented in a digital and/or quantumcomputer program product that includes instructions that are stored onone or more non-transitory machine-readable storage media, and that areexecutable on one or more digital and/or quantum processing devices. Thesystems described in this specification, or portions of them, can eachbe implemented as an apparatus, method, or system that can include oneor more digital and/or quantum processing devices and memory to storeexecutable instructions to perform the operations described in thisspecification.

While this specification contains many specific implementation details,these should not be construed as limitations on the scope of what may beclaimed, but rather as descriptions of features that may be specific toparticular implementations. Certain features that are described in thisspecification in the context of separate implementations can also beimplemented in combination in a single implementation. Conversely,various features that are described in the context of a singleimplementation can also be implemented in multiple implementationsseparately or in any suitable sub-combination. Moreover, althoughfeatures may be described above as acting in certain combinations andeven initially claimed as such, one or more features from a claimedcombination can in some cases be excised from the combination, and theclaimed combination may be directed to a sub-combination or variation ofa sub-combination.

Similarly, while operations are depicted in the drawings in a particularorder, this should not be understood as requiring that such operationsbe performed in the particular order shown or in sequential order, orthat all illustrated operations be performed, to achieve desirableresults. In certain circumstances, multitasking and parallel processingmay be advantageous. Moreover, the separation of various system modulesand components in the implementations described above should not beunderstood as requiring such separation in all implementations, and itshould be understood that the described program components and systemscan generally be integrated together in a single software product orpackaged into multiple software products.

Particular implementations of the subject matter have been described.Other implementations are within the scope of the following claims. Forexample, the actions recited in the claims can be performed in adifferent order and still achieve desirable results. As one example, theprocesses depicted in the accompanying figures do not necessarilyrequire the particular order shown, or sequential order, to achievedesirable results. In some cases, multitasking and parallel processingmay be advantageous.

What is claimed is:
 1. (canceled)
 2. A quantum repeater system forquantum communication, the system comprising: a quantum signal receiverconfigured to receive a quantum field signal comprising one or morequantum analog signals; a quantum signal converter connected to thequantum signal receiver, wherein the quantum signal converter isconfigured to: encode the quantum analog signals as correspondingdigital quantum information; decode the digital quantum information toobtain a recovered quantum field signal; and a quantum signaltransmitter connected to the quantum signal converter, wherein thequantum signal transmitter is configured to transmit the recoveredquantum field signal.
 3. The quantum repeater system of claim 2,wherein: the received quantum field signal comprises a Gottesman, Kitaevand Preskill (GKP) state; the one or more quantum analog signalscomprise one or more GKP state samples; and the recovered quantum fieldsignal comprises a recovered GKP state.
 4. The quantum repeater systemof claim 2, further comprising a quantum memory connected to the quantumsignal converter, wherein the quantum memory is configured to storedigital quantum information encoded by the quantum signal converter, andwherein the quantum memory is further configured to perform quantumerror correction operations on digital quantum information stored in thequantum memory.
 5. The quantum repeater system of claim 4, wherein thequantum error correction operations comprise syndrome measurements,classical decoder operations and quantum error correcting feedbackoperations.
 6. The quantum repeater system of claim 4, wherein thequantum signal converter is further configured to decode error correcteddigital quantum information stored in one or more qudits as a recoveredquantum field signal.
 7. The quantum repeater system of claim 2, furthercomprising multiple quantum signal receivers, multiple quantum signalconverters, and multiple quantum signal transmitters connected to form aquantum network.
 8. The quantum repeater system of claim 2, whereinencoding the quantum analog signals as corresponding digital quantuminformation comprises encoding the quantum analog signals ascorresponding digital quantum information in one or more qudits,comprising, for each sampled quantum analog signal, applying a hybridanalog-digital encoding operation to the quantum analog signal and aqudit in an initial state, wherein the hybrid analog-digital encodingoperation is based on a swap operation comprising three adderoperations.
 9. The quantum repeater system of claim 2, wherein encodingthe quantum analog signals as corresponding digital quantum informationcomprises encoding the quantum analog signals as corresponding digitalquantum information in one or more qudits, comprising, for each sampledquantum analog signal, applying a hybrid analog-digital encodingoperation to the quantum analog signal and a qudit in an initial state,wherein the hybrid analog-digital encoding operation and a correspondinghybrid digital-analog decoding operation comprise: a first unitarytransformation comprising a canonical field momentum operator and aqudit field operator; multiple Fourier transformations; and a secondunitary transformation comprising a canonical field position operatorand the qudit field operator.
 10. The quantum repeater system of claim9, wherein applying the hybrid analog-digital encoding operation to thequantum analog signal and a qudit in an initial state comprises:applying the first unitary transformation to the quantum analog signaland the initial state of the qudit to obtain a first modified quantumanalog signal and a first evolved state of the qudit; sequentiallyapplying two Fourier transformations to the first modified quantumanalog signal to obtain a second modified quantum analog signal;applying a Fourier transformation to the first evolved state of thequdit to obtain a second evolved state of the qudit; applying the secondunitary transformation to the second modified quantum analog signal andthe second evolved state of the qudit to obtain a third modified quantumanalog signal and a third evolved state of the qudit; applying a Fouriertransformation to the third evolved state of the qudit to obtain afourth evolved state of the qudit; and applying the first unitarytransformation to the third modified quantum analog signal and thefourth evolved state of the qudit to obtain a fifth evolved state of thequdit, wherein providing the qudit in the evolved state as a quantumdigital encoding of the quantum analog signal comprises providing thequdit in the fifth evolved state as a quantum digital encoding of thequantum analog signal.
 11. The quantum repeater system of claim 9,wherein applying the hybrid digital-analog decoding operation to thequdit and a quantum analog register in an initial state comprises:sequentially applying two Fourier transformations to the quantum analogregister in the initial state to obtain a first modified state of thequantum analog register; applying a first unitary transformation to thefirst modified state of the quantum analog register and the qudit toobtain a second modified state of the quantum analog register and afirst evolved state of the qudit, wherein the first unitarytransformation comprises a canonical field momentum operator and a quditfield operator; applying a Fourier transformation to the first evolvedstate of the qudit to obtain a second evolved state of the qudit;applying a second unitary transformation to the second modified state ofthe quantum analog register and the second evolved state of the qudit toobtain a third modified state of the quantum analog register and a thirdevolved state of the qudit, wherein the second unitary transformationcomprises a canonical field position operator and the qudit fieldoperator; applying a Fourier transformation to the third evolved stateof the qudit to obtain a fourth evolved state of the qudit; sequentiallyapplying two Fourier transformations to the third modified state of thequantum analog register to obtain a fourth modified state of the quantumanalog register; and applying the first unitary transformation to thefourth modified state of the quantum analog register and the fourthevolved state of the qudit to obtain a fifth modified state of thequantum analog register, wherein providing the modified state of thequantum analog register as a quantum analog encoding of the quantumdigital information comprises providing the fifth modified state of thequantum analog register as the quantum analog encoding of the quantumdigital information.
 12. A method for repeating a quantum field signal,the method comprising: receiving a quantum field signal comprising oneor more quantum analog signals; for each quantum analog signal of theone or more quantum analog signals: encoding the quantum analog signalas corresponding digital quantum information; and decoding the quantumdigital information to obtain a recovered quantum field signal; andtransmitting the recovered quantum field signals.
 13. The method ofclaim 12, wherein: the received quantum field signal comprises aGottesman, Kitaev and Preskill (GKP) state; the one or more quantumanalog signals comprise one or more GKP state samples; and the recoveredquantum field signal comprises a recovered GKP state.
 14. The method ofclaim 12, further comprising storing the digital quantum information ina qudit, wherein storing the corresponding digital quantum informationfurther comprises performing one or more rounds of quantum errorcorrection operations on digital quantum information stored in quantummemory.
 15. The method of claim 14, wherein the quantum error correctionoperations comprise syndrome measurements, classical decoder operationsand quantum error correcting feedback operations.
 16. The method ofclaim 14, wherein decoding the quantum digital information to obtain arecovered quantum field signal comprises, for each qudit storingcorresponding error corrected digital quantum information, decoding theerror corrected quantum digital information by applying a hybriddigital-analog decoding operation to the qudit storing error correcteddigital quantum information and a quantum analog register in an initialstate.
 17. The method of claim 12, wherein encoding the quantum analogsignal as corresponding digital quantum information comprises applying ahybrid analog-digital encoding operation to the quantum analog signaland a qudit in an initial state and decoding the quantum digitalinformation to obtain a recovered quantum field comprises applying ahybrid digital-analog decoding operation to the qudit and a quantumanalog register in an initial state, wherein the hybrid analog-digitalencoding operation are based on a swap operation comprising three adderoperations.
 18. The method of claim 17, wherein the swap operationcomprises: a first adder operation applied to a first signal and asecond signal; two sequential Fourier transformations applied to thesecond signal; a second adder operation applied to the first signal andthe second signal; two sequential Fourier transformations applied to thefirst signal; a third adder operation applied to the first signal andthe second signal; and two sequential Fourier transformations applied tothe second signal.
 19. The method of claim 17, wherein the hybridanalog-digital encoding operation and a corresponding hybriddigital-analog decoding operation comprise: a first unitarytransformation comprising a canonical field momentum operator and aqudit field operator; multiple Fourier transformations; and a secondunitary transformation comprising a canonical field position operatorand the qudit field operator.
 20. The method of claim 19, whereinapplying the hybrid analog-digital encoding operation to the quantumanalog signal and a qudit in an initial state comprises: applying thefirst unitary transformation to the quantum analog signal and theinitial state of the qudit to obtain a first modified quantum analogsignal and a first evolved state of the qudit; sequentially applying twoFourier transformations to the first modified quantum analog signal toobtain a second modified quantum analog signal; applying a Fouriertransformation to the first evolved state of the qudit to obtain asecond evolved state of the qudit; applying the second unitarytransformation to the second modified quantum analog signal and thesecond evolved state of the qudit to obtain a third modified quantumanalog signal and a third evolved state of the qudit; applying a Fouriertransformation to the third evolved state of the qudit to obtain afourth evolved state of the qudit; and applying the first unitarytransformation to the third modified quantum analog signal and thefourth evolved state of the qudit to obtain a fifth evolved state of thequdit, wherein providing the qudit in the evolved state as a quantumdigital encoding of the quantum analog signal comprises providing thequdit in the fifth evolved state as a quantum digital encoding of thequantum analog signal.
 21. The method of claim 19, wherein applying thehybrid digital-analog decoding operation to the qudit and a quantumanalog register in an initial state comprises: sequentially applying twoFourier transformations to the quantum analog register in the initialstate to obtain a first modified state of the quantum analog register;applying a first unitary transformation to the first modified state ofthe quantum analog register and the qudit to obtain a second modifiedstate of the quantum analog register and a first evolved state of thequdit, wherein the first unitary transformation comprises a canonicalfield momentum operator and a qudit field operator; applying a Fouriertransformation to the first evolved state of the qudit to obtain asecond evolved state of the qudit; applying a second unitarytransformation to the second modified state of the quantum analogregister and the second evolved state of the qudit to obtain a thirdmodified state of the quantum analog register and a third evolved stateof the qudit, wherein the second unitary transformation comprises acanonical field position operator and the qudit field operator; applyinga Fourier transformation to the third evolved state of the qudit toobtain a fourth evolved state of the qudit; sequentially applying twoFourier transformations to the third modified state of the quantumanalog register to obtain a fourth modified state of the quantum analogregister; and applying the first unitary transformation to the fourthmodified state of the quantum analog register and the fourth evolvedstate of the qudit to obtain a fifth modified state of the quantumanalog register, wherein providing the modified state of the quantumanalog register as a quantum analog encoding of the quantum digitalinformation comprises providing the fifth modified state of the quantumanalog register as the quantum analog encoding of the quantum digitalinformation.